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%I #21 Nov 16 2022 17:02:09
%S 1,1,-1,-1,0,1,1,-1,0,1,-1,-1,0,1,1,-1,0,1,-1,-1,0,1,1,-1,0,1,-1,-1,0,
%T 1,1,-1,0,1,-1,-1,0,1,1,-1,0,1,-1,-1,0,1,1,-1,0,1,-1,-1,0,1,1,-1,0,1,
%U -1,-1,0,1,1,-1,0,1,-1,-1,0,1,1,-1,0,1,-1,-1,0
%N Expansion of (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)) in powers of x.
%H G. C. Greubel, <a href="/A257196/b257196.txt">Table of n, a(n) for n = 0..2500</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,-1,0,-1).
%F Euler transform of length 10 sequence [1, -2, 0, 0, 1, 0, 0, 1, 0, -1].
%F a(n) is multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4) and a(0) = 1.
%F G.f.: 1 + x / (1 + x^2) - x^2 / (1 + x^4).
%F G.f.: (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)).
%F a(n) = -a(-n) for all n in Z unless n = 0. a(n+8) = a(n) unless n=0 or n=-8. a(4*n) = 0 unless n=0.
%F a(n) = A112299(n) unless n=0. - _R. J. Mathar_, Apr 19 2015
%e G.f. = 1 + x - x^2 - x^3 + x^5 + x^6 - x^7 + x^9 - x^10 - x^11 + x^13 + ...
%t a[ n_] := Boole[n == 0] + {1, -1, -1, 0, 1, 1, -1, 0}[[Mod[ n, 8, 1]]];
%t a[ n_] := If[ n == 0, 1, Sign[ n] SeriesCoefficient[ (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)), {x, 0, Abs @ n}]];
%t CoefficientList[Series[(1 + x)*(1 + x^5)/((1 + x^2)*(1 + x^4)), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 02 2018 *)
%t LinearRecurrence[{0,-1,0,-1,0,-1},{1,1,-1,-1,0,1,1},100] (* _Harvey P. Dale_, Nov 16 2022 *)
%o (PARI) {a(n) = (n==0) + [0, 1, -1, -1, 0, 1, 1, -1][n%8 + 1]};
%o (PARI) {a(n) = if( n==0, 1, n%2, (-1)^(n\2), n%4 == 2, -(-1)^(n\4), 0)};
%o (PARI) {a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)) + x * O(x^abs(n)), abs(n)))};
%o (PARI) x='x+O('x^60); Vec((1 + x)*(1 + x^5)/((1 + x^2)*(1 + x^4))) \\ _G. C. Greubel_, Aug 02 2018
%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x)*(1 + x^5)/((1 + x^2)*(1 + x^4)))); // _G. C. Greubel_, Aug 02 2018
%Y Cf. A112299.
%K sign,mult,easy
%O 0,1
%A _Michael Somos_, Apr 17 2015
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